Ramble, Part 14: Even Stevin

[stoutfellow]

Simon Stevin was a very talented man. Unfortunately, he gets rather lost in the crowd of such people who swarm through the fifteenth through seventeenth centuries; but he deserves to be remembered.

Military history buffs may know him as Maurice of Nassau's quartermaster general; he also devised the system of sluices by means of which the Dutch, in times of dire need, were able to breach the dikes, bringing in the North Sea as a weapon against invading armies (and, not incidentally, allowing the powerful Dutch navy to take a hand). He was a skilled engineer; he designed and built a functioning sail-cart - a wind-propelled land vehicle. But, to the mathematician, he is most noteworthy for another achievement. He did not invent, but he popularized, the decimal point.

The decimal point is a nice illustration of... call it the emergent quality of good notation. Positional notation, as I mentioned before, could naturally be extended to represent arbitrarily large numbers; the realization that it could be extended in the other direction apparently occurred to several people, Stevin among them, in the late sixteenth century. (As often, the Chinese had come up with the idea a good deal earlier.) That development had a number of interesting consequences.

First, it simplified computations involving fractions. Recall that the procedure for adding common fractions - a/b + c/d = (ad + bc)/(bd) - involves three multiplications and an addition, and multiplications are a good deal more difficult and time-consuming than additions. With the introduction of decimal fractions, the computational algorithms of Hindu-Arabic notation extend directly to fractions, and this is a great simplification.

Second, it provided a systematic way of talking about approximations and of evaluating how good they are, in such familiar phrases as "to three digits accuracy". The ancient Greeks struggled with problems of approximation, and achieved such results as that pi lies between 220/71 and 22/7; a modern student can rattle off "three-point-one-four-one-five-nine", with far greater accuracy and a tacit and automatic interval of certainty ("between 3.141585 and 3.141595").

Third, it opened the door to new methods of computation. As mentioned, multiplication is a more difficult operation than addition, and mathematicians and astronomers looked for ways of transforming the former into the latter. The first such way, known as "prosthaphaeresis", involved trigonometry, and required shifting the decimal point to bring the numbers to be multiplied into the range between 0 and 1. (I'll describe the method below.) It was superseded by the discovery of logarithms, but that too required shifting the decimal point.

Prosthaphaeresis rests on a variation of the addition formula for cosine. If A and B are two angles, then cos A cos B = (cos(A+B) + cos(A-B))/2. So, to multiply two numbers X and Y, shift the decimal point to bring them into the range from 0 to 1. Then, locate two angles A and B whose cosines approximately equal those shifted values. (This usually involved table lookup; trigonometric tables began to be widely available at about this time.) Take the sum and difference of A and B, take the cosines of both, and average them; then shift the decimal point back where it belongs.

Fourth, it may have played a role in the developing study of infinite series. Earlier thinkers, from Zeno of Elea to Nicolas Oresme, had done some work in that area, but the fact that as simple a fraction as 1/3 requires an infinite decimal forces infinite series into the heart of mathematics. (How many algebra students have been baffled by the teacher's insistence that 0.999... equals 1?)

None of this, I daresay, was on the minds of Stevin and his contemporaries, but it followed from their work; and that is what good notation is worth.

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Comments

[toraks.livejournal.com]

Cool!

We like decimal points, oh yes we do! ;-p

[hornedhopper.livejournal.com]

"he also devised the system of sluices by means of which the Dutch, in times of dire need, were able to breach the dikes, bringing in the North Sea as a weapon against invading armies (and, not incidentally, allowing the powerful Dutch navy to take a hand). He was a skilled engineer; he designed and built a functioning sail-cart - a wind-propelled land vehicle. But, to the mathematician, he is most noteworthy for another achievement. He did not invent, but he popularized, the decimal point."

Thus illustrating my point made many times before that there are *never* parades for engineers and inventors. I'm sure the general involved got a lot of nice trinkets as a reward. Where are Stevin's publicly acknowledged laurels (outside of military buffs' websites and mathematicians' texts)? His Rose Parade? Is there a day in his honor in Holland? Or here, in Dutch Town, U.S.A.? Ahem, HornedHopper is now stepping off of the soap box she apparently climbed upon, unbeknownst to her. (g)

Thanks for an interesting post

[dan-ad-nauseam.livejournal.com]

It's too bad western mathematics failed to recognize early on that instead of taking additive fractions by going with the largest 1/n possible and continuing from there, using standardized denominators would be more efficient. Had this development happened, we probably would have gone to decimal notation sooner.

[filkferengi]

Cool!